LMFDB - Embedded newform 684.5.h.e.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(37,684)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 588x^{6} + 3710486x^{4} - 3505922196x^{2} + 813472313329 $ x^8 + 588x^6 + 3710486x^4 - 3505922196*x^2 + 813472313329
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.4
Root			$-27.0329 - 37.7795i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.5.h.e.37.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-23.6814 q^{5} +14.3965 q^{7} +O(q^{10})$	$q-23.6814 q^{5} +14.3965 q^{7} +128.461 q^{11} +29.9596i q^{13} -205.987 q^{17} +(-65.5686 - 354.995i) q^{19} +17.2001 q^{23} -64.1895 q^{25} +1139.54i q^{29} +530.233i q^{31} -340.930 q^{35} -1300.14i q^{37} +2709.12i q^{41} +3080.43 q^{43} +26.5899 q^{47} -2193.74 q^{49} -430.049i q^{53} -3042.15 q^{55} -3848.66i q^{59} +843.668 q^{61} -709.487i q^{65} -7060.91i q^{67} -8406.80i q^{71} +3113.84 q^{73} +1849.40 q^{77} -2348.73i q^{79} +3944.83 q^{83} +4878.07 q^{85} -2709.12i q^{89} +431.314i q^{91} +(1552.76 + 8406.80i) q^{95} -11407.9i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 68 q^{7}+O(q^{10}) $ 8 * q - 68 * q^7	$ 8 q - 68 q^{7} + 1124 q^{19} + 36 q^{25} + 5044 q^{43} - 14436 q^{49} - 11332 q^{55} - 10652 q^{61} - 4580 q^{73} - 20140 q^{85}+O(q^{100}) $ 8 * q - 68 * q^7 + 1124 * q^19 + 36 * q^25 + 5044 * q^43 - 14436 * q^49 - 11332 * q^55 - 10652 * q^61 - 4580 * q^73 - 20140 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−23.6814			−0.947257			−0.473629	−	0.880725i	$-0.657056\pi$
$5$	−23.6814			−0.947257			−0.473629	+	0.880725i	$0.657056\pi$
$6$		0			0
$7$	14.3965			0.293806			0.146903	−	0.989151i	$-0.453069\pi$
$7$	14.3965			0.293806			0.146903	+	0.989151i	$0.453069\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	128.461			1.06166			0.530832	−	0.847477i	$-0.321879\pi$
$11$	128.461			1.06166			0.530832	+	0.847477i	$0.321879\pi$
$12$		0			0
$13$			29.9596i			0.177276i	0.996064	+	0.0886380i	$0.0282514\pi$
$13$			29.9596i			0.177276i	−0.996064	+	0.0886380i	$0.971749\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−205.987			−0.712758			−0.356379	−	0.934341i	$-0.615989\pi$
$17$	−205.987			−0.712758			−0.356379	+	0.934341i	$0.615989\pi$
$18$		0			0
$19$	−65.5686	−	354.995i	−0.181630	−	0.983367i
$19$	−65.5686	−	354.995i	−0.181630	−	0.983367i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	17.2001			0.0325143			0.0162572	−	0.999868i	$-0.494825\pi$
$23$	17.2001			0.0325143			0.0162572	+	0.999868i	$0.494825\pi$
$24$		0			0
$25$	−64.1895			−0.102703
$26$		0			0
$27$		0			0
$28$		0			0
$29$			1139.54i			1.35498i	0.735533	+	0.677489i	$0.236932\pi$
$29$			1139.54i			1.35498i	−0.735533	+	0.677489i	$0.763068\pi$
$30$		0			0
$31$			530.233i			0.551751i	0.961193	+	0.275876i	$0.0889678\pi$
$31$			530.233i			0.551751i	−0.961193	+	0.275876i	$0.911032\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−340.930			−0.278310
$36$		0			0
$37$		−	1300.14i		−	0.949703i	−0.880066	−	0.474851i	$-0.842502\pi$
$37$		−	1300.14i		−	0.949703i	0.880066	−	0.474851i	$-0.157498\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2709.12i			1.61161i	0.592179	+	0.805807i	$0.298268\pi$
$41$			2709.12i			1.61161i	−0.592179	+	0.805807i	$0.701732\pi$
$42$		0			0
$43$	3080.43			1.66600			0.832998	−	0.553276i	$-0.186623\pi$
$43$	3080.43			1.66600			0.832998	+	0.553276i	$0.186623\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	26.5899			0.0120371			0.00601854	−	0.999982i	$-0.498084\pi$
$47$	26.5899			0.0120371			0.00601854	+	0.999982i	$0.498084\pi$
$48$		0			0
$49$	−2193.74			−0.913678
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	430.049i		−	0.153097i	−0.997066	−	0.0765484i	$-0.975610\pi$
$53$		−	430.049i		−	0.153097i	0.997066	−	0.0765484i	$-0.0243900\pi$
$54$		0			0
$55$	−3042.15			−1.00567
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	3848.66i		−	1.10562i	−0.833308	−	0.552809i	$-0.813556\pi$
$59$		−	3848.66i		−	1.10562i	0.833308	−	0.552809i	$-0.186444\pi$
$60$		0			0
$61$	843.668			0.226732			0.113366	−	0.993553i	$-0.463837\pi$
$61$	843.668			0.226732			0.113366	+	0.993553i	$0.463837\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	709.487i		−	0.167926i
$66$		0			0
$67$		−	7060.91i		−	1.57294i	−0.617631	−	0.786468i	$-0.711908\pi$
$67$		−	7060.91i		−	1.57294i	0.617631	−	0.786468i	$-0.288092\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8406.80i		−	1.66769i	−0.552002	−	0.833843i	$-0.686136\pi$
$71$		−	8406.80i		−	1.66769i	0.552002	−	0.833843i	$-0.313864\pi$
$72$		0			0
$73$	3113.84			0.584319			0.292160	−	0.956370i	$-0.405626\pi$
$73$	3113.84			0.584319			0.292160	+	0.956370i	$0.405626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1849.40			0.311924
$78$		0			0
$79$		−	2348.73i		−	0.376339i	−0.982137	−	0.188169i	$-0.939745\pi$
$79$		−	2348.73i		−	0.376339i	0.982137	−	0.188169i	$-0.0602554\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3944.83			0.572628			0.286314	−	0.958136i	$-0.407570\pi$
$83$	3944.83			0.572628			0.286314	+	0.958136i	$0.407570\pi$
$84$		0			0
$85$	4878.07			0.675166
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2709.12i		−	0.342018i	−0.985270	−	0.171009i	$-0.945297\pi$
$89$		−	2709.12i		−	0.342018i	0.985270	−	0.171009i	$-0.0547027\pi$
$90$		0			0
$91$			431.314i			0.0520848i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1552.76	+	8406.80i	0.172051	+	0.931502i
$96$		0			0
$97$		−	11407.9i		−	1.21244i	−0.795295	−	0.606222i	$-0.792684\pi$
$97$		−	11407.9i		−	1.21244i	0.795295	−	0.606222i	$-0.207316\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12746.1			−1.24950			−0.624748	−	0.780827i	$-0.714798\pi$
$101$	−12746.1			−1.24950			−0.624748	+	0.780827i	$0.714798\pi$
$102$		0			0
$103$		−	12102.6i		−	1.14079i	−0.821370	−	0.570395i	$-0.806790\pi$
$103$		−	12102.6i		−	1.14079i	0.821370	−	0.570395i	$-0.193210\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5977.12i		−	0.522065i	−0.965330	−	0.261032i	$-0.915937\pi$
$107$		−	5977.12i		−	0.522065i	0.965330	−	0.261032i	$-0.0840629\pi$
$108$		0			0
$109$		−	16111.0i		−	1.35603i	−0.735047	−	0.678017i	$-0.762840\pi$
$109$		−	16111.0i		−	1.35603i	0.735047	−	0.678017i	$-0.237160\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14814.0i			1.16015i	0.814563	+	0.580076i	$0.196977\pi$
$113$			14814.0i			1.16015i	−0.814563	+	0.580076i	$0.803023\pi$
$114$		0			0
$115$	−407.322			−0.0307994
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2965.50			−0.209413
$120$		0			0
$121$	1861.35			0.127132
$122$		0			0
$123$		0			0
$124$		0			0
$125$	16321.0			1.04454
$126$		0			0
$127$		−	14783.8i		−	0.916595i	−0.888799	−	0.458298i	$-0.848459\pi$
$127$		−	14783.8i		−	0.916595i	0.888799	−	0.458298i	$-0.151541\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4624.95			−0.269504			−0.134752	−	0.990879i	$-0.543024\pi$
$131$	−4624.95			−0.269504			−0.134752	+	0.990879i	$0.543024\pi$
$132$		0			0
$133$	−943.958	−	5110.69i	−0.0533641	−	0.288919i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15145.1			0.806920			0.403460	−	0.914997i	$-0.367808\pi$
$137$	15145.1			0.806920			0.403460	+	0.914997i	$0.367808\pi$
$138$		0			0
$139$	9568.91			0.495259			0.247630	−	0.968855i	$-0.420348\pi$
$139$	9568.91			0.495259			0.247630	+	0.968855i	$0.420348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3848.66i			0.188208i
$144$		0			0
$145$		−	26985.9i		−	1.28351i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−32594.6			−1.46816			−0.734079	−	0.679064i	$-0.762386\pi$
$149$	−32594.6			−1.46816			−0.734079	+	0.679064i	$0.762386\pi$
$150$		0			0
$151$		−	29765.9i		−	1.30546i	−0.757589	−	0.652732i	$-0.773623\pi$
$151$		−	29765.9i		−	1.30546i	0.757589	−	0.652732i	$-0.226377\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	12556.7i		−	0.522651i
$156$		0			0
$157$	−19768.3			−0.801993			−0.400996	−	0.916080i	$-0.631336\pi$
$157$	−19768.3			−0.801993			−0.400996	+	0.916080i	$0.631336\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	247.621			0.00955291
$162$		0			0
$163$	31975.8			1.20350			0.601751	−	0.798684i	$-0.294470\pi$
$163$	31975.8			1.20350			0.601751	+	0.798684i	$0.294470\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			26940.6i			0.965994i	0.875622	+	0.482997i	$0.160452\pi$
$167$			26940.6i			0.965994i	−0.875622	+	0.482997i	$0.839548\pi$
$168$		0			0
$169$	27663.4			0.968573
$170$		0			0
$171$		0			0
$172$		0			0
$173$			12126.6i			0.405180i	0.979264	+	0.202590i	$0.0649359\pi$
$173$			12126.6i			0.405180i	−0.979264	+	0.202590i	$0.935064\pi$
$174$		0			0
$175$	−924.105			−0.0301749
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	20963.5i		−	0.654271i	−0.944978	−	0.327135i	$-0.893917\pi$
$179$		−	20963.5i		−	0.654271i	0.944978	−	0.327135i	$-0.106083\pi$
$180$		0			0
$181$		−	18196.3i		−	0.555427i	−0.960664	−	0.277713i	$-0.910423\pi$
$181$		−	18196.3i		−	0.555427i	0.960664	−	0.277713i	$-0.0895765\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			30789.3i			0.899613i
$186$		0			0
$187$	−26461.4			−0.756710
$188$		0			0
$189$		0			0
$190$		0			0
$191$	24755.7			0.678591			0.339295	−	0.940680i	$-0.389811\pi$
$191$	24755.7			0.678591			0.339295	+	0.940680i	$0.389811\pi$
$192$		0			0
$193$		−	45667.2i		−	1.22600i	−0.790084	−	0.612999i	$-0.789963\pi$
$193$		−	45667.2i		−	1.22600i	0.790084	−	0.612999i	$-0.210037\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	29626.2			0.763386			0.381693	−	0.924289i	$-0.375341\pi$
$197$	29626.2			0.763386			0.381693	+	0.924289i	$0.375341\pi$
$198$		0			0
$199$	−15713.8			−0.396804			−0.198402	−	0.980121i	$-0.563575\pi$
$199$	−15713.8			−0.396804			−0.198402	+	0.980121i	$0.563575\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			16405.3i			0.398101i
$204$		0			0
$205$		−	64155.9i		−	1.52661i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8423.03	−	45603.2i	−0.192831	−	1.04401i
$210$		0			0
$211$		−	13987.8i		−	0.314184i	−0.987584	−	0.157092i	$-0.949788\pi$
$211$		−	13987.8i		−	0.314184i	0.987584	−	0.157092i	$-0.0502119\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−72948.9			−1.57813
$216$		0			0
$217$			7633.50i			0.162108i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	6171.30i		−	0.126355i
$222$		0			0
$223$			24181.5i			0.486266i	0.969993	+	0.243133i	$0.0781752\pi$
$223$			24181.5i			0.486266i	−0.969993	+	0.243133i	$0.921825\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	67662.7i		−	1.31310i	−0.754283	−	0.656550i	$-0.772015\pi$
$227$		−	67662.7i		−	1.31310i	0.754283	−	0.656550i	$-0.227985\pi$
$228$		0			0
$229$	52601.6			1.00306			0.501531	−	0.865140i	$-0.332770\pi$
$229$	52601.6			1.00306			0.501531	+	0.865140i	$0.332770\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−58526.5			−1.07805			−0.539027	−	0.842289i	$-0.681208\pi$
$233$	−58526.5			−1.07805			−0.539027	+	0.842289i	$0.681208\pi$
$234$		0			0
$235$	−629.687			−0.0114022
$236$		0			0
$237$		0			0
$238$		0			0
$239$	91511.9			1.60207			0.801036	−	0.598617i	$-0.204283\pi$
$239$	91511.9			1.60207			0.801036	+	0.598617i	$0.204283\pi$
$240$		0			0
$241$		−	18465.4i		−	0.317926i	−0.987285	−	0.158963i	$-0.949185\pi$
$241$		−	18465.4i		−	0.317926i	0.987285	−	0.158963i	$-0.0508150\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	51950.9			0.865488
$246$		0			0
$247$	10635.5	−	1964.41i	0.174327	−	0.0321987i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	69228.6			1.09885			0.549425	−	0.835543i	$-0.314847\pi$
$251$	69228.6			1.09885			0.549425	+	0.835543i	$0.314847\pi$
$252$		0			0
$253$	2209.55			0.0345193
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	33498.4i		−	0.507175i	−0.967312	−	0.253587i	$-0.918389\pi$
$257$		−	33498.4i		−	0.507175i	0.967312	−	0.253587i	$-0.0816105\pi$
$258$		0			0
$259$		−	18717.5i		−	0.279029i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−98446.5			−1.42327			−0.711637	−	0.702547i	$-0.752046\pi$
$263$	−98446.5			−1.42327			−0.711637	+	0.702547i	$0.752046\pi$
$264$		0			0
$265$			10184.2i			0.145022i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			2193.81i			0.0303176i	0.999885	+	0.0151588i	$0.00482538\pi$
$269$			2193.81i			0.0303176i	−0.999885	+	0.0151588i	$0.995175\pi$
$270$		0			0
$271$	65551.4			0.892573			0.446286	−	0.894890i	$-0.352746\pi$
$271$	65551.4			0.892573			0.446286	+	0.894890i	$0.352746\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8245.88			−0.109036
$276$		0			0
$277$	−68404.0			−0.891501			−0.445750	−	0.895157i	$-0.647063\pi$
$277$	−68404.0			−0.891501			−0.445750	+	0.895157i	$0.647063\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	116405.i		−	1.47421i	−0.675778	−	0.737105i	$-0.736192\pi$
$281$		−	116405.i		−	1.47421i	0.675778	−	0.737105i	$-0.263808\pi$
$282$		0			0
$283$	−106110.			−1.32490			−0.662448	−	0.749108i	$-0.730482\pi$
$283$	−106110.			−1.32490			−0.662448	+	0.749108i	$0.730482\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			39001.9i			0.473502i
$288$		0			0
$289$	−41090.3			−0.491976
$290$		0			0
$291$		0			0
$292$		0			0
$293$			126425.i			1.47264i	0.676631	+	0.736322i	$0.263439\pi$
$293$			126425.i			1.47264i	−0.676631	+	0.736322i	$0.736561\pi$
$294$		0			0
$295$			91141.7i			1.04731i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			515.308i			0.00576400i
$300$		0			0
$301$	44347.4			0.489480
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−19979.3			−0.214773
$306$		0			0
$307$			70700.0i			0.750141i	0.926996	+	0.375070i	$0.122381\pi$
$307$			70700.0i			0.750141i	−0.926996	+	0.375070i	$0.877619\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	906.852			0.00937596			0.00468798	−	0.999989i	$-0.498508\pi$
$311$	906.852			0.00937596			0.00468798	+	0.999989i	$0.498508\pi$
$312$		0			0
$313$	81871.5			0.835688			0.417844	−	0.908519i	$-0.362786\pi$
$313$	81871.5			0.835688			0.417844	+	0.908519i	$0.362786\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15460.0i		−	0.153848i	−0.997037	−	0.0769238i	$-0.975490\pi$
$317$		−	15460.0i		−	0.153848i	0.997037	−	0.0769238i	$-0.0245098\pi$
$318$		0			0
$319$			146386.i			1.43853i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13506.3	+	73124.5i	0.129459	+	0.700903i
$324$		0			0
$325$		−	1923.09i		−	0.0182068i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	382.802			0.00353657
$330$		0			0
$331$			48316.1i			0.440997i	0.975387	+	0.220498i	$0.0707684\pi$
$331$			48316.1i			0.440997i	−0.975387	+	0.220498i	$0.929232\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			167212.i			1.48998i
$336$		0			0
$337$		−	53563.1i		−	0.471635i	−0.971797	−	0.235817i	$-0.924223\pi$
$337$		−	53563.1i		−	0.471635i	0.971797	−	0.235817i	$-0.0757767\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			68114.5i			0.585775i
$342$		0			0
$343$	−66148.2			−0.562251
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−219968.			−1.82684			−0.913421	−	0.407016i	$-0.866569\pi$
$347$	−219968.			−1.82684			−0.913421	+	0.407016i	$0.866569\pi$
$348$		0			0
$349$	12409.4			0.101883			0.0509413	−	0.998702i	$-0.483778\pi$
$349$	12409.4			0.101883			0.0509413	+	0.998702i	$0.483778\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	187793.			1.50706			0.753531	−	0.657413i	$-0.228349\pi$
$353$	187793.			1.50706			0.753531	+	0.657413i	$0.228349\pi$
$354$		0			0
$355$			199085.i			1.57973i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	114235.			0.886358			0.443179	−	0.896433i	$-0.353851\pi$
$359$	114235.			0.886358			0.443179	+	0.896433i	$0.353851\pi$
$360$		0			0
$361$	−121723.	+	46553.1i	−0.934021	+	0.357219i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−73740.1			−0.553501
$366$		0			0
$367$	51222.7			0.380303			0.190152	−	0.981755i	$-0.439102\pi$
$367$	51222.7			0.380303			0.190152	+	0.981755i	$0.439102\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	6191.20i		−	0.0449808i
$372$		0			0
$373$			146233.i			1.05106i	0.850774	+	0.525531i	$0.176133\pi$
$373$			146233.i			1.05106i	−0.850774	+	0.525531i	$0.823867\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−34140.1			−0.240205
$378$		0			0
$379$			190003.i			1.32276i	0.750050	+	0.661381i	$0.230029\pi$
$379$			190003.i			1.32276i	−0.750050	+	0.661381i	$0.769971\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			75467.0i			0.514470i	0.966349	+	0.257235i	$0.0828114\pi$
$383$			75467.0i			0.514470i	−0.966349	+	0.257235i	$0.917189\pi$
$384$		0			0
$385$	−43796.4			−0.295472
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−72750.6			−0.480770			−0.240385	−	0.970678i	$-0.577274\pi$
$389$	−72750.6			−0.480770			−0.240385	+	0.970678i	$0.577274\pi$
$390$		0			0
$391$	−3542.99			−0.0231748
$392$		0			0
$393$		0			0
$394$		0			0
$395$			55621.3i			0.356490i
$396$		0			0
$397$	13303.0			0.0844049			0.0422025	−	0.999109i	$-0.486563\pi$
$397$	13303.0			0.0844049			0.0422025	+	0.999109i	$0.486563\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	226296.i		−	1.40730i	−0.710545	−	0.703652i	$-0.751551\pi$
$401$		−	226296.i		−	1.40730i	0.710545	−	0.703652i	$-0.248449\pi$
$402$		0			0
$403$	−15885.6			−0.0978122
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	167018.i		−	1.00827i
$408$		0			0
$409$		−	116364.i		−	0.695620i	−0.937565	−	0.347810i	$-0.886925\pi$
$409$		−	116364.i		−	0.695620i	0.937565	−	0.347810i	$-0.113075\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	55407.2i		−	0.324838i
$414$		0			0
$415$	−93419.3			−0.542426
$416$		0			0
$417$		0			0
$418$		0			0
$419$	116947.			0.666134			0.333067	−	0.942903i	$-0.391916\pi$
$419$	116947.			0.666134			0.333067	+	0.942903i	$0.391916\pi$
$420$		0			0
$421$			104183.i			0.587807i	0.955835	+	0.293903i	$0.0949544\pi$
$421$			104183.i			0.587807i	−0.955835	+	0.293903i	$0.905046\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13222.2			0.0732026
$426$		0			0
$427$	12145.9			0.0666151
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	118318.i		−	0.636934i	−0.947934	−	0.318467i	$-0.896832\pi$
$431$		−	118318.i		−	0.636934i	0.947934	−	0.318467i	$-0.103168\pi$
$432$		0			0
$433$			94677.0i			0.504974i	0.967600	+	0.252487i	$0.0812484\pi$
$433$			94677.0i			0.504974i	−0.967600	+	0.252487i	$0.918752\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1127.78	−	6105.95i	−0.00590558	−	0.0319735i
$438$		0			0
$439$			214927.i			1.11522i	0.830103	+	0.557611i	$0.188282\pi$
$439$			214927.i			1.11522i	−0.830103	+	0.557611i	$0.811718\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	116972.			0.596041			0.298020	−	0.954559i	$-0.403674\pi$
$443$	116972.			0.596041			0.298020	+	0.954559i	$0.403674\pi$
$444$		0			0
$445$			64155.9i			0.323979i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	297312.i		−	1.47475i	−0.675481	−	0.737377i	$-0.736064\pi$
$449$		−	297312.i		−	1.47475i	0.675481	−	0.737377i	$-0.263936\pi$
$450$		0			0
$451$			348018.i			1.71099i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	10214.1i		−	0.0493377i
$456$		0			0
$457$	115224.			0.551711			0.275855	−	0.961199i	$-0.411039\pi$
$457$	115224.			0.551711			0.275855	+	0.961199i	$0.411039\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−308352.			−1.45092			−0.725462	−	0.688262i	$-0.758374\pi$
$461$	−308352.			−1.45092			−0.725462	+	0.688262i	$0.758374\pi$
$462$		0			0
$463$	−73899.9			−0.344732			−0.172366	−	0.985033i	$-0.555141\pi$
$463$	−73899.9			−0.344732			−0.172366	+	0.985033i	$0.555141\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	124904.			0.572720			0.286360	−	0.958122i	$-0.407555\pi$
$467$	124904.			0.572720			0.286360	+	0.958122i	$0.407555\pi$
$468$		0			0
$469$		−	101652.i		−	0.462138i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	395716.			1.76873
$474$		0			0
$475$	4208.81	+	22787.0i	0.0186540	+	0.100995i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−103260.			−0.450051			−0.225026	−	0.974353i	$-0.572247\pi$
$479$	−103260.			−0.450051			−0.225026	+	0.974353i	$0.572247\pi$
$480$		0			0
$481$	38951.8			0.168359
$482$		0			0
$483$		0			0
$484$		0			0
$485$			270155.i			1.14850i
$486$		0			0
$487$			266475.i			1.12357i	0.827284	+	0.561784i	$0.189885\pi$
$487$			266475.i			1.12357i	−0.827284	+	0.561784i	$0.810115\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−58865.7			−0.244174			−0.122087	−	0.992519i	$-0.538959\pi$
$491$	−58865.7			−0.244174			−0.122087	+	0.992519i	$0.538959\pi$
$492$		0			0
$493$		−	234730.i		−	0.965772i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	121029.i		−	0.489976i
$498$		0			0
$499$	−328576.			−1.31958			−0.659789	−	0.751451i	$-0.729354\pi$
$499$	−328576.			−1.31958			−0.659789	+	0.751451i	$0.729354\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−337327.			−1.33326			−0.666630	−	0.745389i	$-0.732264\pi$
$503$	−337327.			−1.33326			−0.666630	+	0.745389i	$0.732264\pi$
$504$		0			0
$505$	301846.			1.18359
$506$		0			0
$507$		0			0
$508$		0			0
$509$			322148.i			1.24342i	0.783246	+	0.621712i	$0.213563\pi$
$509$			322148.i			1.24342i	−0.783246	+	0.621712i	$0.786437\pi$
$510$		0			0
$511$	44828.4			0.171677
$512$		0			0
$513$		0			0
$514$		0			0
$515$			286608.i			1.08062i
$516$		0			0
$517$	3415.78			0.0127793
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	167084.i		−	0.615543i	−0.951460	−	0.307772i	$-0.900417\pi$
$521$		−	167084.i		−	0.615543i	0.951460	−	0.307772i	$-0.0995833\pi$
$522$		0			0
$523$			252855.i			0.924418i	0.886771	+	0.462209i	$0.152943\pi$
$523$			252855.i			0.924418i	−0.886771	+	0.462209i	$0.847057\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	109221.i		−	0.393265i
$528$		0			0
$529$	−279545.			−0.998943
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−81164.3			−0.285700
$534$		0			0
$535$			141547.i			0.494530i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−281811.			−0.970020
$540$		0			0
$541$	−316995.			−1.08307			−0.541536	−	0.840678i	$-0.682157\pi$
$541$	−316995.			−1.08307			−0.541536	+	0.840678i	$0.682157\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			381532.i			1.28451i
$546$		0			0
$547$		−	45420.4i		−	0.151801i	−0.997115	−	0.0759007i	$-0.975817\pi$
$547$		−	45420.4i		−	0.151801i	0.997115	−	0.0759007i	$-0.0241832\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	404530.	−	74717.7i	1.33244	−	0.246105i
$552$		0			0
$553$		−	33813.5i		−	0.110571i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	448596.			1.44592			0.722961	−	0.690889i	$-0.242781\pi$
$557$	448596.			1.44592			0.722961	+	0.690889i	$0.242781\pi$
$558$		0			0
$559$			92288.4i			0.295341i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	284158.i		−	0.896486i	−0.893912	−	0.448243i	$-0.852050\pi$
$563$		−	284158.i		−	0.896486i	0.893912	−	0.448243i	$-0.147950\pi$
$564$		0			0
$565$		−	350816.i		−	1.09896i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			272480.i			0.841608i	0.907152	+	0.420804i	$0.138252\pi$
$569$			272480.i			0.841608i	−0.907152	+	0.420804i	$0.861748\pi$
$570$		0			0
$571$	338225.			1.03737			0.518685	−	0.854965i	$-0.326422\pi$
$571$	338225.			1.03737			0.518685	+	0.854965i	$0.326422\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1104.06			−0.00333932
$576$		0			0
$577$	202048.			0.606882			0.303441	−	0.952850i	$-0.401865\pi$
$577$	202048.			0.606882			0.303441	+	0.952850i	$0.401865\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	56791.8			0.168242
$582$		0			0
$583$		−	55244.7i		−	0.162538i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	104411.			0.303018			0.151509	−	0.988456i	$-0.451587\pi$
$587$	104411.			0.303018			0.151509	+	0.988456i	$0.451587\pi$
$588$		0			0
$589$	188230.	−	34766.6i	0.542574	−	0.100215i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−491121.			−1.39662			−0.698312	−	0.715794i	$-0.746065\pi$
$593$	−491121.			−1.39662			−0.698312	+	0.715794i	$0.746065\pi$
$594$		0			0
$595$	70227.2			0.198368
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	412536.i		−	1.14976i	−0.818237	−	0.574881i	$-0.805048\pi$
$599$		−	412536.i		−	1.14976i	0.818237	−	0.574881i	$-0.194952\pi$
$600$		0			0
$601$		−	680712.i		−	1.88458i	−0.334800	−	0.942289i	$-0.608669\pi$
$601$		−	680712.i		−	1.88458i	0.334800	−	0.942289i	$-0.391331\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−44079.3			−0.120427
$606$		0			0
$607$			239399.i			0.649747i	0.945757	+	0.324874i	$0.105322\pi$
$607$			239399.i			0.649747i	−0.945757	+	0.324874i	$0.894678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			796.624i			0.00213388i
$612$		0			0
$613$	451346.			1.20113			0.600564	−	0.799577i	$-0.294943\pi$
$613$	451346.			1.20113			0.600564	+	0.799577i	$0.294943\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−583146.			−1.53182			−0.765909	−	0.642949i	$-0.777711\pi$
$617$	−583146.			−1.53182			−0.765909	+	0.642949i	$0.777711\pi$
$618$		0			0
$619$	−309282.			−0.807187			−0.403593	−	0.914938i	$-0.632239\pi$
$619$	−309282.			−0.807187			−0.403593	+	0.914938i	$0.632239\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	39001.9i		−	0.100487i
$624$		0			0
$625$	−346386.			−0.886749
$626$		0			0
$627$		0			0
$628$		0			0
$629$			267813.i			0.676909i
$630$		0			0
$631$	−213923.			−0.537277			−0.268638	−	0.963241i	$-0.586574\pi$
$631$	−213923.			−0.537277			−0.268638	+	0.963241i	$0.586574\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			350101.i			0.868252i
$636$		0			0
$637$		−	65723.7i		−	0.161973i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			20918.0i			0.0509102i	0.999676	+	0.0254551i	$0.00810349\pi$
$641$			20918.0i			0.0509102i	−0.999676	+	0.0254551i	$0.991897\pi$
$642$		0			0
$643$	356385.			0.861979			0.430990	−	0.902357i	$-0.358165\pi$
$643$	356385.			0.861979			0.430990	+	0.902357i	$0.358165\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	725461.			1.73303			0.866514	−	0.499152i	$-0.166355\pi$
$647$	725461.			1.73303			0.866514	+	0.499152i	$0.166355\pi$
$648$		0			0
$649$		−	494404.i		−	1.17380i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−69815.2			−0.163728			−0.0818642	−	0.996643i	$-0.526087\pi$
$653$	−69815.2			−0.163728			−0.0818642	+	0.996643i	$0.526087\pi$
$654$		0			0
$655$	109525.			0.255289
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	260250.i		−	0.599266i	−0.954055	−	0.299633i	$-0.903136\pi$
$659$		−	260250.i		−	0.599266i	0.954055	−	0.299633i	$-0.0968643\pi$
$660$		0			0
$661$		−	541390.i		−	1.23910i	−0.784956	−	0.619551i	$-0.787315\pi$
$661$		−	541390.i		−	1.23910i	0.784956	−	0.619551i	$-0.212685\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	22354.3	+	121029.i	0.0505496	+	0.273681i
$666$		0			0
$667$			19600.1i			0.0440562i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	108379.			0.240713
$672$		0			0
$673$		−	733045.i		−	1.61845i	−0.587496	−	0.809227i	$-0.699886\pi$
$673$		−	733045.i		−	1.61845i	0.587496	−	0.809227i	$-0.300114\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			729503.i			1.59166i	0.605521	+	0.795829i	$0.292965\pi$
$677$			729503.i			1.59166i	−0.605521	+	0.795829i	$0.707035\pi$
$678$		0			0
$679$		−	164234.i		−	0.356224i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	277766.i		−	0.595439i	−0.954653	−	0.297719i	$-0.903774\pi$
$683$		−	277766.i		−	0.595439i	0.954653	−	0.297719i	$-0.0962259\pi$
$684$		0			0
$685$	−358657.			−0.764361
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12884.1			0.0271404
$690$		0			0
$691$	206537.			0.432555			0.216277	−	0.976332i	$-0.430608\pi$
$691$	206537.			0.432555			0.216277	+	0.976332i	$0.430608\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−226605.			−0.469138
$696$		0			0
$697$		−	558044.i		−	1.14869i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	682169.			1.38821			0.694106	−	0.719872i	$-0.255800\pi$
$701$	682169.			1.38821			0.694106	+	0.719872i	$0.255800\pi$
$702$		0			0
$703$	−461545.	+	85248.5i	−0.933906	+	0.172495i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−183499.			−0.367110
$708$		0			0
$709$	542363.			1.07894			0.539471	−	0.842004i	$-0.318624\pi$
$709$	542363.			1.07894			0.539471	+	0.842004i	$0.318624\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			9120.05i			0.0179398i
$714$		0			0
$715$		−	91141.7i		−	0.178281i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−533539.			−1.03207			−0.516035	−	0.856568i	$-0.672592\pi$
$719$	−533539.			−1.03207			−0.516035	+	0.856568i	$0.672592\pi$
$720$		0			0
$721$		−	174236.i		−	0.335171i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	73146.3i		−	0.139161i
$726$		0			0
$727$	25159.5			0.0476029			0.0238014	−	0.999717i	$-0.492423\pi$
$727$	25159.5			0.0476029			0.0238014	+	0.999717i	$0.492423\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−634528.			−1.18745
$732$		0			0
$733$	−257793.			−0.479804			−0.239902	−	0.970797i	$-0.577115\pi$
$733$	−257793.			−0.479804			−0.239902	+	0.970797i	$0.577115\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	907055.i		−	1.66993i
$738$		0			0
$739$	−113196.			−0.207273			−0.103637	−	0.994615i	$-0.533048\pi$
$739$	−113196.			−0.207273			−0.103637	+	0.994615i	$0.533048\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	999125.i		−	1.80985i	−0.425573	−	0.904924i	$-0.639927\pi$
$743$		−	999125.i		−	1.80985i	0.425573	−	0.904924i	$-0.360073\pi$
$744$		0			0
$745$	771886.			1.39072
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	86049.6i		−	0.153386i
$750$		0			0
$751$		−	467170.i		−	0.828315i	−0.910205	−	0.414157i	$-0.864076\pi$
$751$		−	467170.i		−	0.828315i	0.910205	−	0.414157i	$-0.135924\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			704899.i			1.23661i
$756$		0			0
$757$	255923.			0.446599			0.223300	−	0.974750i	$-0.428317\pi$
$757$	255923.			0.446599			0.223300	+	0.974750i	$0.428317\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	214533.			0.370445			0.185223	−	0.982697i	$-0.440699\pi$
$761$	214533.			0.370445			0.185223	+	0.982697i	$0.440699\pi$
$762$		0			0
$763$		−	231943.i		−	0.398411i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	115304.			0.196000
$768$		0			0
$769$	385015.			0.651066			0.325533	−	0.945531i	$-0.394456\pi$
$769$	385015.			0.651066			0.325533	+	0.945531i	$0.394456\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	590817.i		−	0.988767i	−0.869244	−	0.494383i	$-0.835394\pi$
$773$		−	590817.i		−	0.988767i	0.869244	−	0.494383i	$-0.164606\pi$
$774$		0			0
$775$		−	34035.4i		−	0.0566667i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	961726.	−	177633.i	1.58481	−	0.292718i
$780$		0			0
$781$		−	1.07995e6i		−	1.77052i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	468142.			0.759693
$786$		0			0
$787$			248476.i			0.401176i	0.979676	+	0.200588i	$0.0642853\pi$
$787$			248476.i			0.401176i	−0.979676	+	0.200588i	$0.935715\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			213269.i			0.340860i
$792$		0			0
$793$			25276.0i			0.0401940i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	868055.i		−	1.36657i	−0.730154	−	0.683283i	$-0.760552\pi$
$797$		−	868055.i		−	1.36657i	0.730154	−	0.683283i	$-0.239448\pi$
$798$		0			0
$799$	−5477.18			−0.00857953
$800$		0			0
$801$		0			0
$802$		0			0
$803$	400008.			0.620351
$804$		0			0
$805$	−5864.02			−0.00904906
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6609.93			−0.0100995			−0.00504975	−	0.999987i	$-0.501607\pi$
$809$	−6609.93			−0.0100995			−0.00504975	+	0.999987i	$0.501607\pi$
$810$		0			0
$811$			404192.i			0.614534i	0.951623	+	0.307267i	$0.0994144\pi$
$811$			404192.i			0.614534i	−0.951623	+	0.307267i	$0.900586\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−757234.			−1.14003
$816$		0			0
$817$	−201979.	−	1.09354e6i	−0.302595	−	1.63829i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−702475.			−1.04218			−0.521092	−	0.853501i	$-0.674475\pi$
$821$	−702475.			−1.04218			−0.521092	+	0.853501i	$0.674475\pi$
$822$		0			0
$823$	−131870.			−0.194691			−0.0973453	−	0.995251i	$-0.531035\pi$
$823$	−131870.			−0.194691			−0.0973453	+	0.995251i	$0.531035\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			475359.i			0.695041i	0.937672	+	0.347521i	$0.112976\pi$
$827$			475359.i			0.695041i	−0.937672	+	0.347521i	$0.887024\pi$
$828$		0			0
$829$			421720.i			0.613642i	0.951767	+	0.306821i	$0.0992653\pi$
$829$			421720.i			0.613642i	−0.951767	+	0.306821i	$0.900735\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	451882.			0.651231
$834$		0			0
$835$		−	637992.i		−	0.915045i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			561175.i			0.797213i	0.917122	+	0.398606i	$0.130506\pi$
$839$			561175.i			0.797213i	−0.917122	+	0.398606i	$0.869494\pi$
$840$		0			0
$841$	−591262.			−0.835964
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−655110.			−0.917488
$846$		0			0
$847$	26796.9			0.0373523
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	22362.6i		−	0.0308789i
$852$		0			0
$853$	−64887.4			−0.0891789			−0.0445894	−	0.999005i	$-0.514198\pi$
$853$	−64887.4			−0.0891789			−0.0445894	+	0.999005i	$0.514198\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			385446.i			0.524810i	0.964958	+	0.262405i	$0.0845157\pi$
$857$			385446.i			0.524810i	−0.964958	+	0.262405i	$0.915484\pi$
$858$		0			0
$859$	337128.			0.456887			0.228443	−	0.973557i	$-0.426636\pi$
$859$	337128.			0.456887			0.228443	+	0.973557i	$0.426636\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			683119.i			0.917223i	0.888637	+	0.458612i	$0.151653\pi$
$863$			683119.i			0.917223i	−0.888637	+	0.458612i	$0.848347\pi$
$864$		0			0
$865$		−	287176.i		−	0.383810i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	301721.i		−	0.399546i
$870$		0			0
$871$	211542.			0.278844
$872$		0			0
$873$		0			0
$874$		0			0
$875$	234965.			0.306894
$876$		0			0
$877$		−	274124.i		−	0.356408i	−0.983994	−	0.178204i	$-0.942971\pi$
$877$		−	274124.i		−	0.356408i	0.983994	−	0.178204i	$-0.0570287\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−173055.			−0.222963			−0.111481	−	0.993767i	$-0.535560\pi$
$881$	−173055.			−0.222963			−0.111481	+	0.993767i	$0.535560\pi$
$882$		0			0
$883$	−796974.			−1.02217			−0.511085	−	0.859530i	$-0.670756\pi$
$883$	−796974.			−1.02217			−0.511085	+	0.859530i	$0.670756\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			990564.i			1.25903i	0.776989	+	0.629514i	$0.216746\pi$
$887$			990564.i			1.25903i	−0.776989	+	0.629514i	$0.783254\pi$
$888$		0			0
$889$		−	212835.i		−	0.269301i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1743.46	−	9439.30i	−0.00218630	−	0.0118369i
$894$		0			0
$895$			496445.i			0.619763i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−604220.			−0.747611
$900$		0			0
$901$			88584.6i			0.109121i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			430915.i			0.526132i
$906$		0			0
$907$		−	716223.i		−	0.870630i	−0.900278	−	0.435315i	$-0.856637\pi$
$907$		−	716223.i		−	0.870630i	0.900278	−	0.435315i	$-0.143363\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			981340.i			1.18245i	0.806506	+	0.591225i	$0.201356\pi$
$911$			981340.i			1.18245i	−0.806506	+	0.591225i	$0.798644\pi$
$912$		0			0
$913$	506759.			0.607939
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−66583.1			−0.0791818
$918$		0			0
$919$	1.13064e6			1.33873			0.669363	−	0.742935i	$-0.266567\pi$
$919$	1.13064e6			1.33873			0.669363	+	0.742935i	$0.266567\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	251865.			0.295640
$924$		0			0
$925$			83455.6i			0.0975376i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	561785.			0.650937			0.325469	−	0.945553i	$-0.394478\pi$
$929$	561785.			0.650937			0.325469	+	0.945553i	$0.394478\pi$
$930$		0			0
$931$	143840.	+	778768.i	0.165952	+	0.898481i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	626644.			0.716800
$936$		0			0
$937$	38883.3			0.0442878			0.0221439	−	0.999755i	$-0.492951\pi$
$937$	38883.3			0.0442878			0.0221439	+	0.999755i	$0.492951\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	726249.i		−	0.820175i	−0.912046	−	0.410087i	$-0.865498\pi$
$941$		−	726249.i		−	0.820175i	0.912046	−	0.410087i	$-0.134502\pi$
$942$		0			0
$943$			46597.1i			0.0524005i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.06681e6			−1.18956			−0.594782	−	0.803887i	$-0.702762\pi$
$947$	−1.06681e6			−1.18956			−0.594782	+	0.803887i	$0.702762\pi$
$948$		0			0
$949$			93289.4i			0.103586i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	166485.i		−	0.183311i	−0.995791	−	0.0916556i	$-0.970784\pi$
$953$		−	166485.i		−	0.183311i	0.995791	−	0.0916556i	$-0.0292159\pi$
$954$		0			0
$955$	−586250.			−0.642800
$956$		0			0
$957$		0			0
$958$		0			0
$959$	218036.			0.237078
$960$		0			0
$961$	642374.			0.695570
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1.08147e6i			1.16134i
$966$		0			0
$967$	−61871.2			−0.0661661			−0.0330830	−	0.999453i	$-0.510533\pi$
$967$	−61871.2			−0.0661661			−0.0330830	+	0.999453i	$0.510533\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.16302e6i			1.23353i	0.787147	+	0.616766i	$0.211557\pi$
$971$			1.16302e6i			1.23353i	−0.787147	+	0.616766i	$0.788443\pi$
$972$		0			0
$973$	137759.			0.145510
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	638654.i		−	0.669077i	−0.942382	−	0.334539i	$-0.891420\pi$
$977$		−	638654.i		−	0.669077i	0.942382	−	0.334539i	$-0.108580\pi$
$978$		0			0
$979$		−	348018.i		−	0.363108i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1.02064e6i		−	1.05624i	−0.849169	−	0.528122i	$-0.822897\pi$
$983$		−	1.02064e6i		−	1.05624i	0.849169	−	0.528122i	$-0.177103\pi$
$984$		0			0
$985$	−701592.			−0.723123
$986$		0			0
$987$		0			0
$988$		0			0
$989$	52983.5			0.0541687
$990$		0			0
$991$			1.16402e6i			1.18525i	0.805477	+	0.592627i	$0.201909\pi$
$991$			1.16402e6i			1.18525i	−0.805477	+	0.592627i	$0.798091\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	372127.			0.375876
$996$		0			0
$997$	−1.71138e6			−1.72170			−0.860848	−	0.508862i	$-0.830066\pi$
$997$	−1.71138e6			−1.72170			−0.860848	+	0.508862i	$0.830066\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.h.e.37.4	yes	8
3.2	odd	2	inner	684.5.h.e.37.6	yes	8
19.18	odd	2	inner	684.5.h.e.37.3	✓	8
57.56	even	2	inner	684.5.h.e.37.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.5.h.e.37.3	✓	8	19.18	odd	2	inner
684.5.h.e.37.4	yes	8	1.1	even	1	trivial
684.5.h.e.37.5	yes	8	57.56	even	2	inner
684.5.h.e.37.6	yes	8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-23.6814 q^{5} +14.3965 q^{7} +O(q^{10})\)	\(q-23.6814 q^{5} +14.3965 q^{7} +128.461 q^{11} +29.9596i q^{13} -205.987 q^{17} +(-65.5686 - 354.995i) q^{19} +17.2001 q^{23} -64.1895 q^{25} +1139.54i q^{29} +530.233i q^{31} -340.930 q^{35} -1300.14i q^{37} +2709.12i q^{41} +3080.43 q^{43} +26.5899 q^{47} -2193.74 q^{49} -430.049i q^{53} -3042.15 q^{55} -3848.66i q^{59} +843.668 q^{61} -709.487i q^{65} -7060.91i q^{67} -8406.80i q^{71} +3113.84 q^{73} +1849.40 q^{77} -2348.73i q^{79} +3944.83 q^{83} +4878.07 q^{85} -2709.12i q^{89} +431.314i q^{91} +(1552.76 + 8406.80i) q^{95} -11407.9i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 68 q^{7}+O(q^{10}) \) 8 * q - 68 * q^7	\( 8 q - 68 q^{7} + 1124 q^{19} + 36 q^{25} + 5044 q^{43} - 14436 q^{49} - 11332 q^{55} - 10652 q^{61} - 4580 q^{73} - 20140 q^{85}+O(q^{100}) \) 8 * q - 68 * q^7 + 1124 * q^19 + 36 * q^25 + 5044 * q^43 - 14436 * q^49 - 11332 * q^55 - 10652 * q^61 - 4580 * q^73 - 20140 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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